Towards a Stable Numerical Evolution of Strongly Gravitating Systems in General Relativity: The Conformal Treatments

We study the stability of three-dimensional numerical evolutions of the Einstein equations, comparing the standard ADM formulation to variations on a family of formulations that separate out the conformal and traceless parts of the system. We develop an implementation of the conformal-traceless (CT) approach that has improved stability properties in evolving weak and strong gravitational fields, and for both vacuum and spacetimes with active coupling to matter sources. Cases studied include weak and strong gravitational wave packets, black holes, boson stars and neutron stars. We show under what conditions the CT approach gives better results in 3D numerical evolutions compared to the ADM formulation. In particular, we show that our implementation of the CT approach gives more long term stable evolutions than ADM in all the cases studied, but is less accurate in the short term for the range of resolutions used in our 3D simulations.Comment: 17 pages, 15 figures. Small changes in the text, and a change in the list of authors. One new reference adde

Numerical hydrodynamics in general relativity

The current status of numerical solutions for the equations of ideal general relativistic hydrodynamics is reviewed. Different formulations of the equations are presented, with special mention to conservative and hyperbolic formulations well-adapted to advanced numerical methods. A representative sample of available numerical schemes is discussed and particular emphasis is paid on solution procedures based on schemes exploiting the characteristic structure of the equations through linearized Riemann solvers. A comprehensive summary of relevant astrophysical simulations in strong gravitational fields, including gravitational collapse, accretion onto black holes and evolution of neutron stars, is also presented. (orig.)SIGLEAvailable from TIB Hannover: RR 4697(1240) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Nonlinear r-modes in rapidly rotating relativistic stars

SIGLEAvailable from: http://www.mpa-garching.mpg.de / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Relativistic simulations of rotational core collapse. II. Collapse dynamics and gravitational radiation

SIGLEAvailable from: http://www.mpa-garching.mpg.de / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Analysis of relativistic hydrodynamics in conservation form

Formulations of Eulerian general relativistic ideal hydrodynamics in conservation form are analyzed in some detail, with particular emphasis to geometric source terms. Simple linear transformations of the equations are introduced and the associated equivalence class is exploited for the optimization of such sources. A significant reduction of their complexity is readily possible in generic spacetimes. The local characteristic structure of the standard member of the equivalence class is analyzed for a general equation of state (EOS). This extends previous results restricted to the polytropic case. The properties of all other members of the class, in particular specialized forms employing Killing symmetries, are derivable from the standard form. Special classes of EOS are identified for both spacelike and null foliations, which lead to explicit inversion of the state vector and computational savings. The entire approach is equally applicable to spacelike or lightlike foliations and presents a complete proposal for numerical relativistic hydrodynamics on stationary or dynamic geometries. (orig.)SIGLEAvailable from TIB Hannover: RR 4697(1248) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Axisymmetric modes of rotating relativistic stars in the cowling approximation

Axisymmetric pulsations of rotating neutron stars can be excited in several scenarios, such as core-collapse, crust and core-quakes and binary mergers and could become detectable either in gravitational waves or high-energy radiation. Here, we present a comprehensive study of all low-order axisymmetric modes of uniformly and rapidly rotating relativistic stars. Initial stationary configurations are appropriately perturbed and are numerically evolved using an axisymmetric, nonlinear relativistic hydrodynamics code, assuming time-independence of the gravitational field (Cowling approximation). The simulations are performed using a high-resolution shock-capturing finite-difference scheme accurate enough to maintain the initial rotation law for a large number of rotational periods, even for stars at the mass-shedding limit. Through Fourier transforms of the time evolution of selected fluid variables, we compute the frequencies of quasi-radial and non-radial modes with spherical harmonic indices l=0, 1, 2 and 3, for a sequence of rotating stars from the non-rotating limit to the mass-shedding limit. The frequencies of the axisymmetric modes are affected significantly by rotation only when the rotation rate exceeds about 50% of the maximum allowed. As expected, at large rotation rates, apparent mode crossings between different modes appear. In addition to the above modes, several axisymmetric inertial modes are also excited in our numerical evolutions. (orig.)SIGLEAvailable from Internet: http://www.mpa-garching.mpg.de / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Morphology and dynamics of highly supersonic relativistic jets

We present a simulation of a diffuse (#eta#=0.01), high beam Mach number (M_b=6.0), relativistic (beam Lorentz factor 22.4) axisymmetric jet, and discuss its morphology and dynamics. The jet exhibits a prominent structure of oblique shocks inside the beam and possesses a dominant thick cocoon. This result is qualitatively different from the findings of other recent simulations of low beam Mach number relativistic jets, where both features are absent. We find that the jet propagates very efficiently through the ambient medium. Its mean velocity is fifteen times larger than that expected from classical (i.e. nonrelativistic) simulations. The simulations are performed with a high-resolution shock-capturing scheme using a Riemann solver which is based on the spectral decomposition of the Jacobian matrices of relativistic hydrodynamics. (orig.)Available from TIB Hannover: RR 4697(869) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Morphology and dynamics of relativistic jets

We present a comprehensive analysis of the morphology and dynamics of relativistic pressure-matched axisymmetric jets. The numerical simulations have been carried out with a high-resolution shock-capturing hydro-code based on an aprpoximate relativistic Riemann solver derived form the spectral decomposition of the Jacobian matrices of relativistic hydrodynamics. We discuss the dependence of the jet morphology on several parameters paying special attention to the relativistic effects caused by high Lorentz factors and large internal energies of the beam flow. The parameter space of our analysis is spanned by the ratio of the beam and ambient medium rest mass density (#eta#), the beam Mach number (M_b), the beam Lorentz factor (W_b) and the adiabatic index (#gamma#) of the equation of state (assuming an ideal gas). Both the ultrarelativistic regime (W_b#>=#20) and the hypersonic regime (relativistic March number greater than 100) have been studied. Our results show that the enhancement of the effective inertial mass of the beam due to relativistic effects (through the specific enthalpy and the Lorentz factor) makes relativistic jets significantly more stable than Newtonian jets. We find that relativistic jets propagate very efficiently through the ambient medium at speeds, which agree very well with those obtained form an estimate based on a one-dimensional momentum balance. The propagation efficiency of a relativistic jet is an increasing function of the beam flow velocity. The cocoons of relativistic jets are in general less prominent than those of Newtonian jets. For models with #gamma#=5/3 the cocoon is mainly formed by large vortices and has an almost constant thickness along the beam. Jets with #gamma#=4/3 have thinner cocoons with smaller vortices. For jets with high beam internal energies the cocoons reduce to a small lobe near the head of the jet. The cocoon of a highly supersonic jet is extended and overpressured. Its thickness decreases and the mean pressure inside the cocoon increases with increasing beam flow velocity or beam Lorentz factor. (orig.)Available from TIB Hannover: RR 4697(895) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman